Fundamentals of Plasma Physics

3.9 Wave-particle energy transfer 111

speed, andvfas the fast speed (out of traffic jam). It is tempting, but wrong, to say that the

average velocity is(1−α)vf+αvsbecause

average velocity of a trip=

total distance

total time

. (3.204)

Since the fast-portion duration istf= (1−α)L/vfwhile the the slow-portion duration is

ts=αL/vs,the average velocity of the complete trip is

vavg=

L

(1−α)L/vf+αL/vs

=

1

(1−α)/vf+α/vs

. (3.205)

Thus, ifvs<< vfthenvavg≃vs/αwhich (i) is not the weighted average of the fast and

slow velocities and (ii) is almost entirely determined by the slow velocity.

3.9.2 Motion of particles in a sawtooth potential

The exact motion of a particle in a sinusoidal potential can be solved using elliptic integrals,

but the obtained solution is implicit, i.e., the solution is expressed in the form of time

as a function of position. While exact, the implicit nature of this solution obscures the

essential physics. In order to shed some light on the underlying physics, we willfirst

consider particle motion in the contrived, but analytically tractable, situation of the periodic

sawtooth-shaped potential shown in Fig.3.17 and then later will consider particle motion in

a more natural, but harder to analyze, sinusoidal potential.

When in the downward-sloping portion of the sawtooth potential, a particleexperiences

a constant acceleration+aand when in the upward portion it experiences a constant accel-

eration−a.Our goal is to determine the average velocity of a group of particles injected

with an initial velocityv 0 into the system. Care is required when using the word ‘average’

because this word has two meanings depending on whether one is referring to the average

velocity of a single particle or the average velocity of a group of particles. The average ve-

locity of a single particle is defined by Eq.(3.204) whereas the average velocity of a group

of particles is defined as the sum of the velocities of all the particles in the group divided

by the number of particles in the group.

The average velocity of any given individual particle depends on where the particle

was injected. Consider the four particles denoted asA,B, C,andDin Fig.3.17 as rep-

resentatives of the various possibilities for injection location. ParticleAis injected at a

potential maximum, particleCat a potential minimum, particleBis injected half way on

the downslope, and particleDis injected half way on the upslope.

B

A

C

D

Figure 3.17: Initial positions of particlesA,B,C, andD. All are injected with same initial
velocityv 0 , moving to the right.